Final answer:
The question is about solving a set of trigonometric equations for values of x within the range [0, 2π). The solutions involve using basic trigonometric identities and properties to find the angles where the sine and cosine functions are equal or have matching values.
Step-by-step explanation:
The student is asking for solutions to a series of trigonometric equations with the solutions restricted to lying within the range [0, 2π). These equations involve the sine and cosine functions. Let's solve each part of the question step by step:
a) sin(2x) = cos(x)
To solve sin(2x) = cos(x), recall that cos(x) can be written as sin(90 degrees - x) or sin(π/2 - x). This allows us to set up the equation as sin(2x) = sin(π/2 - x). By the arcsin function, we find that 2x and (π/2 - x) must be equal to some point where their sine values match, leading to potential solutions for x.
b) sin(2x) = sin(x)
Here, we use the property that if sin(a) = sin(b), then a = b + k2π or a = π - b + k2π, where k is an integer. Applying these conditions allows us to find the values of x that satisfy the original equation.
c) sin(x) = cos(x)
Similar to part a), rewrite cos(x) as sin(π/2 - x) to get sin(x) = sin(π/2 - x). This implies x = π/4 or 3π/4, the angles in the first and second quadrants where the sine and cosine values are equal.
Unfortunately, the given references provided do not offer direct methods to solve these specific equations, but the trigonometric identities and properties are vital in obtaining the correct solutions.
Complete question version:
Solve the following equation, giving the exact solutions which lie in [0, 2π). (Enter your answers as a comma-separated list.)
a.) sin(2x) = cos(x)
b.) sin(2x) = sin(x)
c.) sin(x) = cos(x)